Are simple animal snares and traps a form of automation? Of computation?
Absolutely these traps and snares are a form of automation.  


They take a task--harvesting small animals--which was traditionally done by hunting them, and make the process automatic.  The mechanism requires a human to set up, but its function is automatic.  This is to say that the mechanism operates without human involvement.


Absolutely this is a form of computation.


As DuttaA observed, these machines utilize a simple "IF/THEN" statement.  In the case of the snare:


  IF the hook is displaced from the base, THEN the sapling straightens


These simple machines will also return True or False:


  TRUE: The trap catches an animal 
  FALSE: The trap is sprung but empty


The small animals are the input and, potentially, the output, depending on whether the mechanism returns "true".  

(The use of "true" has historically included phrases such as "their aim was true" in the sense of shooting an arrow or throwing a spear.)